How do you determine if arcsin(sin(-2π/7))=-2π/7?
in general
arcsin(sinx) = x , since the operations are inverses of each other
However, since arcsinØ has no unique answer, it is common practise to state the solution closest to zero
e.g. suppose you do arcsin(sin 200°) on your calculator
sin 200° = -.342...
and arcsin(200°)
= arcsin (-.342..) = -20° , which is correct since sin(-20) = -.342...
Also using the CAST rule, we know sin 200° = sin(-20°) or -sin20°
Since -2π/7 is in quadrant IV (appr -64°)
arcsin(sin(-2π/7)) = -2π/7
try it on your calculator, make sure you set it to radians
To determine if arcsin(sin(-2π/7)) = -2π/7, we need to understand the definitions of arcsin and sin, as well as their relationship in the given equation.
1. The sin function (short for sine) is a trigonometric function that represents the ratio of the length of the side opposite to an angle in a right triangle to the length of the hypotenuse. It takes an angle as input and returns a value between -1 and 1.
2. The arcsin function (short for inverse sine or sin^(-1)) is the inverse of the sin function. It takes a value between -1 and 1 as input and returns the angle whose sine is equal to that value. The output of arcsin is usually expressed in radians.
Now, let's determine if arcsin(sin(-2π/7)) equals -2π/7 by following these steps:
Step 1: Evaluate sin(-2π/7)
Given the input -2π/7, we need to calculate the sine function. The sine function will take the input angle (-2π/7) and return a value between -1 and 1.
Step 2: Calculate arcsin(sin(-2π/7))
Take the result from Step 1 (the value of sin(-2π/7)) and apply the arcsin function to it. This will give us the angle whose sin is equal to that value.
Step 3: Compare the result to -2π/7
Finally, check if the result obtained in Step 2 is equal to -2π/7.
By following these steps, we can determine if arcsin(sin(-2π/7)) equals -2π/7.